Theorem coss12d

Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019) (Proof shortened by BJ, 8-Nov-2026)

	Ref	Expression
Hypotheses	coss12d.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	coss12d.c	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )
Assertion	coss12d	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		coss12d.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		coss12d.c	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )
3		coss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) )
5		coss2	⊢ ( 𝐶 ⊆ 𝐷 → ( 𝐵 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐷 ) )
6	2 5	syl	⊢ ( 𝜑 → ( 𝐵 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐷 ) )
7	4 6	sstrd	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐷 ) )